1 Overview

Goal: Describe variation in MFSR Chinook Salmon spawn timing (i.e., putative redd completion date), and how it relates to environmental covariates.

Objectives:

  1. compile data on spawn timing, stream temperature, stream flow, elevation, and slope for Chinook Salmon in the Middle Fork Salmon River (MFSR) from 2002 to 2005.

  2. compute several summary statistics and data visualizations to examine variation in MFSR Chinook Salmon spawn timing.

  3. fit a series of linear mixed models (LMMs) to test for relationships between spawn timing (day of year) and environmental covariates.

1.1 Study Area

This study was conducted in the Middle Fork of the Salmon River (MFSR) in central Idaho (Fig. 1). The MFSR is a tributary of the Salmon River and is part of the larger Columbia River Basin. The MFSR is home to several species of salmon, including Chinook salmon (Oncorhynchus tshawytscha).

Map of the Middle Fork Salmon River watershed showing major tributaries and surveyed Chinoon Salmon redd locations from 2002 to 2005.

Figure 1: Map of the Middle Fork Salmon River watershed showing major tributaries and surveyed Chinoon Salmon redd locations from 2002 to 2005.

2 Datasets

2.1 Spawn timing data

Spawn timing data for Chinook salmon were collected from 2001 to 2005 in the MFSR. We removed data from 2001, and data from Knapp Creek and Cape Horn Creek, as these sites were not consistently sampled. Because redds were not observed daily, we inferred spawn dates as the initial date (day of year) a completed redd was observed.

We spatially joined each redd GPS location to the NHDPlus Version 2 (Horizon Systems, 2018) to assign stream reaches based on a common identifier (COMID). The COMID is used to link redd data with covariate data associated with the stream reach on which it is located. A summary of the dataset is shown in Table 1.

Table 1: Data summary
Name spawn_data
Number of rows 3016
Number of columns 6
_______________________
Column type frequency:
Date 1
factor 4
numeric 1
________________________
Group variables None

Variable type: Date

skim_variable n_missing complete_rate min max median n_unique
spawn_date 0 1 2002-08-02 2005-09-11 2003-08-22 115

Variable type: factor

skim_variable n_missing complete_rate ordered n_unique top_counts
redd_id 0 1 FALSE 3016 129: 1, 129: 1, 129: 1, 129: 1
COMID 0 1 FALSE 104 235: 216, 235: 200, 235: 144, 235: 125
stream 0 1 FALSE 8 Big: 647, Bea: 479, Elk: 471, Mar: 368
year 0 1 FALSE 4 200: 1217, 200: 1187, 200: 404, 200: 208

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100
yday 0 1 239.7 8.36 206 234 241 246 263

2.1.1 Descriptive statistics and vizualizations

Figure 2 provides a visual summary of the distribution of spawn timing (day of year, yday), and variation by stream system and year.

Spawn timing is generally unimodal, with a peak in late August to early September. The distribution is slightly left skewed, but the mean and median are similar, indicating low skewness. The variance is low, suggesting no overdispersion. Suggests Poisson family for response if count of spawning events per day, or Gaussian if modeling day of year as continuous.

There is also substantial variation in spawn date by stream and year (Fig. 2B-C; Fig. 3) and within COMIDs within streams (Fig. 4).

Histogram and density of spawn timing for all streams and years (A), and by stream (B) and year (C). In panel (A), colored, vertical lines indicate the mean spawn date for each year., and the black vertical line indicated the global mean.

Figure 2: Histogram and density of spawn timing for all streams and years (A), and by stream (B) and year (C). In panel (A), colored, vertical lines indicate the mean spawn date for each year., and the black vertical line indicated the global mean.

Spawning phenology of adult Chinook Salmon. In all panels, the black density function represented stream-level spawn timing, while the colored density functions represent the spawn timing of individual years. The dashed vertical purple lines represent the 5th and 95th percentiles of the basin-wide spawn timing.

Figure 3: Spawning phenology of adult Chinook Salmon. In all panels, the black density function represented stream-level spawn timing, while the colored density functions represent the spawn timing of individual years. The dashed vertical purple lines represent the 5th and 95th percentiles of the basin-wide spawn timing.

Distribution of spawn time variation by COMID and stream.

Figure 4: Distribution of spawn time variation by COMID and stream.

The temporal progression of Chinook salmon spawning activity varied within each stream across the four study years (Figure 5). The cumulative proportion of redds provides an intuitive measure of the spawning season’s pace and timing. Streams like Marsh and Sulphur exhibit rapid increases, suggesting short, concentrated spawning windows. In contrast, streams like Big and Camas show more gradual increases, indicating a more protracted spawning season. Year-to-year variation is evident in several streams. Overall, this figure highlights both spatial and temporal heterogeneity in spawn timing that underpins the need for models incorporating stream- and year-specific effects.

Proportion of cumulative Chinook salmon redds over time (day of year) across years (2002–2005) and streams. Each line represents a different year, with color denoting the year. Stream-specific panels illustrate temporal variation in the progression of spawning activity, as measured by cumulative redd counts normalized to the maximum value in each stream-year combination.

Figure 5: Proportion of cumulative Chinook salmon redds over time (day of year) across years (2002–2005) and streams. Each line represents a different year, with color denoting the year. Stream-specific panels illustrate temporal variation in the progression of spawning activity, as measured by cumulative redd counts normalized to the maximum value in each stream-year combination.

2.1.2 Evaluate Grouping structure

The structure of the data is repeated measures on COMIDs, and multiple years, shared across COMIDs within streams.

  • Redds observed at specific locations, COMID
  • Nested within stream
  • Observed across multiple years

Table 2 shows the number of COMIDs per stream, and Figure 6 shows the number of observations (unique redds) per COMID.

Many COMIDs only have 1-2 observation, so the groups are not well sampled. There are 23 COMIDs with <5 redds (26%), 13 with <= 2. (12.5%). With <5 obs/level, variance estimates can become unstable and lead to overfitting and absorbing noise (low AIC / high R2) and singular fits. Something to keep in mind during model fitting.

Table 2: Number of COMIDs per stream (major tributaries).
stream n_COMIDs
Big 21
Loon 19
Camas 16
Marsh 13
Bear Valley 11
Sulphur 11
Beaver 7
Elk 6
Number of observations (redds) per COMID.

Figure 6: Number of observations (redds) per COMID.

2.2 Covariates

To test for environmental factors driving variation in spawn timing, we quantified associations with metrics describing thermal and physical conditions in stream reaches. We selected and interrogated covariates based on the following criteria: (1) they are known to influence spawn timing, (2) they are available for all streams, and (3) they are not highly correlated with each other.

Our initial, focal independent variables are:

  • stream temperature (°C)
  • stream discharge (cms)
  • elevation (m above sea level)
  • stream gradient (slope)

2.2.1 Stream temperature

We used modeled daily average stream temperature values predicted at the stream segment (COMID) scale (Siegel et al. 2023). These data were downloaded and filtered to 2002-2005 and for the MFSR. Figure 7 shows the modeled thermal regimes for MFSR tributaries.

Modeled thermal regimes (2001-2005) for MFSR tributaries.

Figure 7: Modeled thermal regimes (2001-2005) for MFSR tributaries.

We calculated metrics relative to a COMID a redd was constructed on and a redd completion date; before, after, and spanning the date. For example, temp_30_before is the average temperature for a COMID where a redd was constructed calculated over the previous 30 days. We did this for 30, 60, and 90 days. We also calculated a time invariant metric relative to a fixed date across all years that was chosen to represent an initial spawning window, e.g., August 1.

The time invariant and after metrics were omitted from further consideration as preliminary data exploration showed weak if any relationship with spawn timing.

2.2.2 Discharge (streamflow)

Stream flow data were compiled from a single USGS Gage lower in the watershed (MF Salmon River at MF Lodge NR Yellow Pine ID - 13309220; Figure 8).

Inter-annual variability in daily discharge (cfs) at MF Lodge USGS Gage 13309220.

Figure 8: Inter-annual variability in daily discharge (cfs) at MF Lodge USGS Gage 13309220.

We calculated flow metrics relative to a COMID a redd was constructed on and a redd completion date; before, after, and spanning the date.For example, temp_30_before is the average temperature for a COMID where a redd was constructed calculated over the previous 30 days. We did this for 30, 60, and 90 days.

The spanning and after metrics were omitted from further consideration as preliminary data exploration showed weak if any relationship with spawn timing.

2.2.3 Elevation and stream gradient (slope)

Elevation and stream slope data were available at the COMID (stream reach) scale from the NHD.

3 Exploratory Data Analysis

3.1 Bivariate relationships

We did exploratory data analysis on the compiled dataset to identify candidate covariates for inclusion in model selection.

Bivariate relationships between spawn date (yday) and continuous predictors (Figure 9), and pairwise correlations between all continuous covariates (Figure 10), are shown below.

Bivariate relationships between spawn date (`yday`) and continuous covariates. Solid lines are linear fits.

Figure 9: Bivariate relationships between spawn date (yday) and continuous covariates. Solid lines are linear fits.

Pairwise correlations between continuous covariates.

Figure 10: Pairwise correlations between continuous covariates.

Takeaways from Figures 9 and 10:

Amoung temperature variables, temp_90 clearly shows the strongest relationship with spawn date (Figure 9), and is highly colinear with temp_30 and temp_60 (Figure 10).

The is a weak negative relationship between mean_elevation and yday (Figure 9), and it is weakly correlated with temp_90 (0.31, Figure 10).

  • flow_30 = decaying exponential, obvious year effect
  • flow_60 = ditto
  • flow_90 = inflections, year effect clear
  • slope = no relationship

3.2 temp_90

Figure 11 shows the relationship between temp_90 and spawn date by stream and year.

Relationship between `temp_90` and `yday` (spawn date) by stream and year. Solid lines are linear fits.

Figure 11: Relationship between temp_90 and yday (spawn date) by stream and year. Solid lines are linear fits.

We evaluated the role of temp_90 in explaining variation in Chinook salmon spawn timing:

m.t90.1 <- lmer(yday ~ (1|COMID), data = df_mod, REML = FALSE)
m.t90.2 <- lmer(yday ~ temp_90 + (1|COMID), data = df_mod, REML = FALSE)
m.t90.3 <- lmer(yday ~ temp_90 + I(temp_90^2) + (1|COMID), data = df_mod, REML = FALSE)

The model with a quadratic effect of temp_90 (m.t90.3) improved dramatically over the null (intercept-only) model and linear effect (Table 3), and boosting both conditional and marginal R² (Table 4). This confirms a strong, nonlinear effect of pre-spawn temperature on spawn timing (Figure 12).

Table 3: AIC comparison of linear mixed models relating spawn date (yday) to temp_90.
Model df AIC delta
m.t90.3 5 15848.81 0.0000
m.t90.2 4 16570.12 721.3107
m.t90.1 3 18629.30 2780.4987
Table 4: Model performance of linear mixed models relating spawn date (yday) to temp_90.
Name Model R2_conditional R2_marginal ICC RMSE AIC_wt BIC_wt Performance_Score
m.t90.3 lmerMod 0.9177407 0.7643117 0.6509825 3.108315 1 1 0.8333333
m.t90.2 lmerMod 0.9086448 0.6915320 0.7038423 3.489863 0 0 0.6100606
m.t90.1 lmerMod 0.6571743 0.0000000 0.6571743 4.929494 0 0 0.0195227
Predicted marginal effect of `temp_90` on `yday` (spawn date) for (A) `m.t90.2` and (B) `m.t90.3`.

Figure 12: Predicted marginal effect of temp_90 on yday (spawn date) for (A) m.t90.2 and (B) m.t90.3.

Next, we add in stream and year as main effects, and then interactions between temp_90 and stream and year:

m.t90.4 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + (1|COMID), data = df_mod, REML = FALSE)
m.t90.5 <- lmer(yday ~ temp_90 + I(temp_90^2) + year + (1|COMID), data = df_mod, REML = FALSE)
m.t90.6 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + (1|COMID), data = df_mod, REML = FALSE)
m.t90.7 <- lmer(yday ~ temp_90 * stream + I(temp_90^2) + year + (1|COMID), data = df_mod, REML = FALSE)
m.t90.8 <- lmer(yday ~ temp_90 * year + I(temp_90^2) + stream + (1|COMID), data = df_mod, REML = FALSE)

Table 5 shows that adding stream (m.t90.4) or year (m.t90.5) individually improved model fit relative to m.t90.3, but year alone (m.t90.5) performed better than stream alone (m.t90.4).

Combining stream and year additively (m.t90.6) provided further improvement in AIC, along with the best marginal R2 and RMSE (Table 6). This suggests that both stream and year provide meaningful structure in explaining variation in spawn timing.

Adding the interaction between temperature and stream (m.t90.7) increased complexity, slightly lowered the marginal R2, and has high multicollinearity (VIF ~15).

Table 5: AIC comparison of linear mixed models relating spawn date (yday) to temp_90.
Model df AIC delta
m.t90.8 18 12136.83 0.0000
m.t90.7 22 13038.01 901.1783
m.t90.6 15 13473.12 1336.2837
m.t90.5 8 13531.17 1394.3321
m.t90.4 12 15807.98 3671.1454
m.t90.3 5 15848.81 3711.9726
Table 6: Model performance of linear mixed models relating spawn date (yday) to temp_90.
Name Model R2_conditional R2_marginal ICC RMSE AIC_wt BIC_wt Performance_Score
m.t90.8 lmerMod 0.9928189 0.6448836 0.9797782 1.581472 1 1 0.8333333
m.t90.7 lmerMod 0.9852284 0.7177821 0.9476590 1.865263 0 0 0.5267941
m.t90.6 lmerMod 0.9800601 0.7293886 0.9263154 2.022117 0 0 0.5061439
m.t90.5 lmerMod 0.9871140 0.6600787 0.9620912 2.021778 0 0 0.4528690
m.t90.3 lmerMod 0.9177407 0.7643117 0.6509825 3.108315 0 0 0.2196386
m.t90.4 lmerMod 0.8982595 0.7921275 0.5105629 3.110542 0 0 0.1666667

Adding the interaction between temperature and year (m.t90.8) gave the lowest AIC overall but had the lowest marginal R² of all models tested. This means that while the model “fit” the data better on paper (AIC), it did so by overfitting or by capturing year-to-year idiosyncrasies that may not generalize well to new data (suspicious marginal effect, Figure 13).

Predicted marginal effect (`m.t90.8`) of `temp_90` on `yday` (accounting for the effects of stream and year).

Figure 13: Predicted marginal effect (m.t90.8) of temp_90 on yday (accounting for the effects of stream and year).

The suspicious curvature in the marginal effect plot of m.t90.8 (showing an unrealistic predicted relationship between temp_90 and yday) confirms the risk of confounding or overfitting with interaction terms—particularly given that the random intercepts already capture substantial site-level variation.

Given the trade-offs, model m.t90.6 emerged as the best compromise: it retained additive main effects of temperature, stream, and year, had high conditional and marginal R², moderate AIC, low RMSE, and avoided high multicollinearity.

3.2.1 Summary

The exploratory analysis of temperature effects on spawn timing revealed a clear, nonlinear relationship between pre-spawn temperature and the day of year when spawning occurs. A quadratic term for temperature consistently improved model fit over linear-only specifications. When adding stream and year as main effects, model performance improved substantially, with both factors accounting for additional variation in spawn timing. However, introducing interaction terms between temperature and either stream or year led to either inflated multicollinearity (VIFs > 10) or implausible model predictions, indicating potential overfitting. Although the temperature-by-year interaction model (m.t90.8) had the lowest AIC, it yielded a counterintuitive relationship between temperature and spawn timing. Therefore, the additive model with quadratic temperature and main effects of stream and year (m.t90.6) was selected as the preferred model, balancing goodness-of-fit with interpretability and biological realism.

3.3 mean_elevation

Figure 14 shows the relationship between between (A) mean_elevation and temp_90 by stream and (B) mean_elevation and yday (spawn date) by stream and year.

Relationship between (A) `mean_elevation` and `temp_90` by stream and (B) `mean_elevation` and `yday` (spawn date) by stream and year. Solid lines are linear fits.

Figure 14: Relationship between (A) mean_elevation and temp_90 by stream and (B) mean_elevation and yday (spawn date) by stream and year. Solid lines are linear fits.

We evaluated the role of mean elevation in explaining variation in Chinook salmon spawn timing:

m.ele.1 <- lmer(yday ~ (1|COMID), data = df_mod, REML = FALSE)
m.ele.2 <- lmer(yday ~ mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m.ele.3 <- lmer(yday ~ mean_elevation + stream + (1|COMID), data = df_mod, REML = FALSE)
m.ele.4 <- lmer(yday ~ mean_elevation + year + (1|COMID), data = df_mod, REML = FALSE)
m.ele.5 <- lmer(yday ~ mean_elevation + stream + year + (1|COMID), data = df_mod, REML = FALSE)
m.ele.6 <- lmer(yday ~ mean_elevation * stream + (1|COMID), data = df_mod, REML = FALSE)
m.ele.7 <- lmer(yday ~ mean_elevation * year + (1|COMID), data = df_mod, REML = FALSE)

Including mean elevation as a main effect improved model fit (ΔAIC = 19 between m.ele.1 and m.ele.2). The best-supported model (m.ele.5) included mean elevation alongside stream and year (ΔAIC = 281 between m.ele.2 and m.ele.5), with moderate collinearity (VIFs ~6), suggesting that mean elevation can contribute to explaining spawn timing when controlling for stream and year (15).

However, adding interactions with stream or year substantially increased model complexity and collinearity, as evidenced by extremely high VIFs (>45). Given the known inverse relationship between elevation and temperature across streams (Figure 14A), caution is warranted when later incorporating both elevation and temperature in the same model.

Table 7: AIC comparison of linear mixed models relating spawn date (yday) to mean_elevation.
Model df AIC delta
m.ele.5 14 18329.29 0.00000
m.ele.6 18 18371.62 42.33148
m.ele.3 11 18431.88 102.58771
m.ele.7 10 18435.12 105.82567
m.ele.4 7 18501.44 172.14764
m.ele.2 4 18611.27 281.97710
m.ele.1 3 18629.30 300.01337
Table 8: Model performance of linear mixed models relating spawn date (yday) to mean_elevation.
Name Model R2_conditional R2_marginal ICC RMSE Sigma AIC_wt AICc_wt BIC_wt Performance_Score
m.ele.5 lmerMod 0.6399737 0.5939968 0.1132428 4.876512 4.927506 1 1 1 0.6843974
m.ele.7 lmerMod 0.6684326 0.1243275 0.6213568 4.774852 4.851946 0 0 0 0.5175646
m.ele.4 lmerMod 0.6574860 0.1155940 0.6127186 4.835441 4.913296 0 0 0 0.4006749
m.ele.1 lmerMod 0.6571743 0.0000000 0.6571743 4.929494 5.009972 0 0 0 0.2596896
m.ele.2 lmerMod 0.6503692 0.1014398 0.6108988 4.929830 5.009158 0 0 0 0.2505245
m.ele.6 lmerMod 0.6398941 0.6301923 0.0262347 4.989054 5.016809 0 0 0 0.1678027
m.ele.3 lmerMod 0.6269782 0.5838335 0.1036716 4.971609 5.022058 0 0 0 0.1413263
Table 9: Variance inflation factors for model m.ele.5.
GVIF Df GVIF^(1/(2*Df))
mean_elevation 5.739424 1 2.395709
stream 5.943076 7 1.135760
year 1.042077 3 1.006893
Table 9: Variance inflation factors for model m.ele.6.
GVIF Df GVIF^(1/(2*Df))
mean_elevation 2.317261e+03 1 48.13794
stream 8.381552e+25 7 71.06668
mean_elevation:stream 5.223825e+25 7 68.70672
Estimated relationship between spawn date and mean elevation (m.ele.5).

Figure 15: Estimated relationship between spawn date and mean elevation (m.ele.5).

3.4 slope

slope is not related to yday (Figure 16A), though there is some association between slope and yday when considering stream (Figure 16B). We will likely drop slope.

Bivariate relationship between `slope` and spawn date (A) and by stream (B). Solid lines are linear fits.

Figure 16: Bivariate relationship between slope and spawn date (A) and by stream (B). Solid lines are linear fits.

3.5 flow_90

Because flow data are not COMID- or stream-specific, it makes sense to think about and represent flow as an out-of-basin year effect that determines when adults make it back to the MFSR and initially onto the spawning grounds.

The strong correlation between flow_90 and year can be seen clearly in (Figure 17A).

Spawn timing vs. 90-day mean discharge pre spawn at MF Lodge.

Figure 17: Spawn timing vs. 90-day mean discharge pre spawn at MF Lodge.

Next we compare the following simple linear models to examine functional structure:

# flow_90 models
m1 <- lm(yday ~ flow_90, data = model_data)
m2 <- lm(yday ~ I(flow_90^2), data = model_data)
m3 <- lm(yday ~ flow_90 + year, data = model_data)
m4 <- lm(yday ~ flow_90 + year + stream, data = model_data)
m5 <- lm(yday ~ flow_90 * stream, data = model_data)
m6 <- lm(yday ~ flow_90 * stream + year, data = model_data)
m7 <- lm(yday ~ flow_90 * year + stream, data = model_data)
m8 <- lm(yday ~ flow_90 * stream + I(flow_90^2), data = model_data)
m9 <- lm(yday ~ flow_90 * stream + year + I(flow_90^2), data = model_data)

AIC scores suggest m7 is best model (Table 10). However, while the R^2 value is 0.98, the parameter estimate for flow_90 has is incredibly small and most variation is now attributed to the year contrasts (Table 11). Thus, flow_90 is clearly confounded with year, and confirmed by high Variance Inflation Factor (VIF) scores (Figure 18).

Table 10: Model performance of linear models relating spawn date (yday) to flow_90.
Name Model R2 R2_adjusted RMSE Sigma AIC_wt AICc_wt BIC_wt Performance_Score
m7 lm 0.9780529 0.9779505 1.237710 1.240800 1 1 1 1.0000000
m9 lm 0.9407030 0.9403269 2.034449 2.041229 0 0 0 0.4960534
m6 lm 0.9108990 0.9103638 2.493858 2.501751 0 0 0 0.4472891
m4 lm 0.8913935 0.8909958 2.753330 2.758824 0 0 0 0.4181935
m3 lm 0.8745347 0.8743681 2.959322 2.961778 0 0 0 0.3942842
m8 lm 0.7348812 0.7334668 4.301804 4.313980 0 0 0 0.2174139
m5 lm 0.7130109 0.7115760 4.475722 4.487641 0 0 0 0.1921569
m1 lm 0.5900974 0.5899614 5.348977 5.350752 0 0 0 0.0576227
m2 lm 0.5352432 0.5350890 5.695650 5.697539 0 0 0 0.0000000
Table 11: Parameter estimates for m7, yday ~ flow_90 * year + stream.
Parameter Coefficient SE CI CI_low CI_high t df_error p
(Intercept) 278.2566486 0.2130585 0.95 277.8388932 278.6744041 1306.0106434 3001 0.0000000
flow_90 -0.0219390 0.0001136 0.95 -0.0221617 -0.0217163 -193.1695196 3001 0.0000000
year2003 -8.3607844 0.2371172 0.95 -8.8257131 -7.8958557 -35.2601350 3001 0.0000000
year2004 11.0240656 0.3856186 0.95 10.2679620 11.7801692 28.5879998 3001 0.0000000
year2005 1.0123700 0.7619753 0.95 -0.4816766 2.5064167 1.3286127 3001 0.1840768
streamBeaver 0.5591186 0.1048594 0.95 0.3535151 0.7647220 5.3320808 3001 0.0000001
streamBig -0.4395848 0.0772852 0.95 -0.5911220 -0.2880475 -5.6878290 3001 0.0000000
streamCamas -0.0934454 0.0899470 0.95 -0.2698094 0.0829186 -1.0388944 3001 0.2989375
streamElk 0.2219142 0.0861007 0.95 0.0530918 0.3907365 2.5773800 3001 0.0100026
streamLoon 0.0895360 0.1073905 0.95 -0.1210304 0.3001024 0.8337428 3001 0.4044923
streamMarsh -0.7143977 0.0913025 0.95 -0.8934196 -0.5353759 -7.8245141 3001 0.0000000
streamSulphur -0.0665471 0.1017872 0.95 -0.2661269 0.1330327 -0.6537861 3001 0.5132997
flow_90:year2003 0.0094506 0.0001190 0.95 0.0092172 0.0096840 79.3900831 3001 0.0000000
flow_90:year2004 -0.0101387 0.0002436 0.95 -0.0106163 -0.0096610 -41.6197601 3001 0.0000000
flow_90:year2005 -0.0046105 0.0005766 0.95 -0.0057411 -0.0034799 -7.9957348 3001 0.0000000
Variance inflation factors (VIF) for model `m7`, `yday ~ flow_90 * year + stream`.

Figure 18: Variance inflation factors (VIF) for model m7, yday ~ flow_90 * year + stream.

3.5.1 Why flow_90 is problematic

Not spatially resolved

  • We are modeling spawn timing at the redd level (COMID/stream)
  • But flow_90 is calculated from a single downstream gauge, and applied to all redds
  • This assumes flow conditions are identical across all sites
  • Including it gives the illusion of spatially resolved variation that isn’t there

Correlated with year

  • Since flow_90 varies mostly across years, it is strongly confounded with year
  • Any flow-related signal is probably already captured by your year fixed effect
  • Including both flow_90 and year risks collinearity, and may produce misleading inferences

Spawn-time aligned flow ≠ experienced flow

  • While flow_90 is aligned to each redd’s spawn date, it still reflects a lower-basin gauge, not the actual hydrologic conditions experienced at the redd site
  • So it might be precisely wrong — aligned in time but irrelevant in space

Recommendation:

Drop flow_90 from model.

Although we initially considered including 90-day mean streamflow (flow_90) as a predictor of spawn timing, this variable was ultimately excluded due to concerns about ecological validity and model overfitting. Stream flow data were derived from a single downstream USGS gauge and did not capture spatial variation across the study streams or reaches. Moreover, because flow_90 was closely aligned with year, it introduced strong collinearity with the year effect and risked attributing site-level variation to flow patterns not actually experienced by individual redds. As such, we excluded flow_90 to avoid misleading inference.

4 Model fitting

4.1 Final dataset

The final dataset includes:

  • yday: spawn date, continuous response variable
  • comid, stream, year: grouping variables
  • temp_90: 90-day mean temperature pre-spawn, continuous predictor variable
  • slope and mean_elevation: provisionally retaining continuous predictor variable

We scaled the continuous covariates to have a mean of 0 and standard deviation of 1. This is important for mixed models, as it helps with convergence and interpretation (Table 12).

Table 12: Final dataset for modeling, first 5 rows.
yday COMID stream year temp_90_raw mean_elevation_raw slope_raw temp_90 mean_elevation slope
235 23519365 Bear Valley 2002 11.07830 1951.730 0.0029916 -0.3178564 0.7181159 -0.5189141
235 23519365 Bear Valley 2002 11.07830 1951.730 0.0029916 -0.3178564 0.7181159 -0.5189141
235 23519319 Bear Valley 2002 12.15472 1946.735 0.0017181 0.4665480 0.6954808 -0.7107062
235 23519319 Bear Valley 2002 12.15472 1946.735 0.0017181 0.4665480 0.6954808 -0.7107062
235 23519319 Bear Valley 2002 12.15472 1946.735 0.0017181 0.4665480 0.6954808 -0.7107062

4.2 Model specification

Variable Fixed? Random? Why
COMID No Yes Not estimating COMID effects, just accounting for correlation, repeated measures
Stream Yes Maybe 8 streams to compare
Year Yes Maybe 4 years to compare

In this case, we certainly will need at least COMID as a random effect to account for the repeated measures. We would like stream as a fixed effect to compare average effects across streams, and year as a fixed effect to compare average effects across years. Though the later maybe be better included as a random effect, as it is not a treatment effect, but rather a random sample of years.

In mixed-effects models, it’s generally recommended to determine the fixed effects structure before deciding on the random effects structure. This approach helps ensure that information intended for fixed effects isn’t unintentionally absorbed by random effects.

4.3 Simple linear models

We conducted a comprehensive brute-force model selection exercise, fitting 31 additive linear models that combined various combinations of predictors: temp_90, stream, year, mean_elevation, and slope:

# All combinations of 1–5 fixed effects: 31 total
m1 <- lm(yday ~ temp_90, data = df_mod)
m2 <- lm(yday ~ stream, data = df_mod)
m3 <- lm(yday ~ year, data = df_mod)
m4 <- lm(yday ~ mean_elevation, data = df_mod)
m5 <- lm(yday ~ slope, data = df_mod)

m6  <- lm(yday ~ temp_90 + stream, data = df_mod)
m7  <- lm(yday ~ temp_90 + year, data = df_mod)
m8  <- lm(yday ~ temp_90 + mean_elevation, data = df_mod)
m9  <- lm(yday ~ temp_90 + slope, data = df_mod)
m10 <- lm(yday ~ stream + year, data = df_mod)
m11 <- lm(yday ~ stream + mean_elevation, data = df_mod)
m12 <- lm(yday ~ stream + slope, data = df_mod)
m13 <- lm(yday ~ year + mean_elevation, data = df_mod)
m14 <- lm(yday ~ year + slope, data = df_mod)
m15 <- lm(yday ~ mean_elevation + slope, data = df_mod)

m16 <- lm(yday ~ temp_90 + stream + year, data = df_mod)
m17 <- lm(yday ~ temp_90 + stream + mean_elevation, data = df_mod)
m18 <- lm(yday ~ temp_90 + stream + slope, data = df_mod)
m19 <- lm(yday ~ temp_90 + year + mean_elevation, data = df_mod)
m20 <- lm(yday ~ temp_90 + year + slope, data = df_mod)
m21 <- lm(yday ~ temp_90 + mean_elevation + slope, data = df_mod)
m22 <- lm(yday ~ stream + year + mean_elevation, data = df_mod)
m23 <- lm(yday ~ stream + year + slope, data = df_mod)
m24 <- lm(yday ~ stream + mean_elevation + slope, data = df_mod)
m25 <- lm(yday ~ year + mean_elevation + slope, data = df_mod)

m26 <- lm(yday ~ temp_90 + stream + year + mean_elevation, data = df_mod)
m27 <- lm(yday ~ temp_90 + stream + year + slope, data = df_mod)
m28 <- lm(yday ~ temp_90 + stream + mean_elevation + slope, data = df_mod)
m29 <- lm(yday ~ temp_90 + year + mean_elevation + slope, data = df_mod)
m30 <- lm(yday ~ stream + year + mean_elevation + slope, data = df_mod)

m31 <- lm(yday ~ temp_90 + stream + year + mean_elevation + slope, data = df_mod)

Overall, the best-fitting model was m31 (temp_90 + stream + year + mean_elevation + slope), though its improvement in AIC (ΔAIC = 82.3) and R² (0.838) over simpler models was modest (Table 13 and 14).

Models excluding elevation and/or slope (e.g., m16: temp_90 + stream + year) performed nearly as well (R² = 0.816), indicating that these terms contributed little to model explanatory power.

Table 13: AIC selection for additive linear models; top 5 of 31 shown.
Model df AIC delta
m31 15 15907.01 0.00000
m26 14 15989.35 82.33843
m27 14 16231.24 324.23284
m16 13 16285.23 378.22148
m28 12 16982.41 1075.40461
Table 14: Model performance for additive linear models.
Name Model R2 R2_adjusted RMSE AIC_wt BIC_wt Performance_Score
m31 lm 0.8378549 0.8371527 3.364204 1 1 1.0000000
m26 lm 0.8332567 0.8325904 3.411572 0 0 0.5959131
m27 lm 0.8193324 0.8186104 3.551163 0 0 0.5836523
m16 lm 0.8159471 0.8152732 3.584278 0 0 0.5807192
m28 lm 0.7679280 0.7671557 4.024776 0 0 0.5400984

Although adding elevation (e.g., m26) improved AIC compared to the base model (m16), the effect estimate for elevation was consistently opposite to ecological expectations—higher elevations predicted later spawn dates (Figure 19 B)) despite the observed negative relationship between elevation and both temperature and spawn timing for several streams (Figure 14). This suggests potential collinearity or confounding, likely due to the inverse relationship between temperature and elevation among streams.

Predicted relationship (red line) between spawn date (`yday`) and (A) `temp_90`, (B) `mean_elevation`, and `slope` for the top additive linear model (`m31`).

Figure 19: Predicted relationship (red line) between spawn date (yday) and (A) temp_90, (B) mean_elevation, and slope for the top additive linear model (m31).

Finally, slope consistently exhibited minimal impact on model performance, and its effect estimate remained small and sometimes non-significant. Thus, we conclude that temp_90, stream, and year are the most consistent predictors of spawn timing.

4.3.1 Summary

Based on model parsimony, interpretability, and consistency with ecological understanding, we selected m16 (temp_90 + stream + year) as the best baseline model to advance into mixed-effects modeling, acknowledging that elevation might be considered cautiously in future hierarchical models if appropriate.

4.4 Targeted model comparison

We will now compare the best base additive model (m16, yday ~ temp_90 + stream + year) to a series of models with increasing complexity to evaluate the contribution of each fixed effect and random effect structure.

4.4.1 Random intercepts

We will start with a random intercept model to account for the repeated measures on COMIDs. This is a baseline mixed model that will be compared to the additive linear model (m16). We will also compare to model m26 (elevation included as a fixed effect) with COMID random intercepts to see if it improves performance.

m16_re <- lmer(yday ~ temp_90 + stream + year + (1 | COMID), data = df_mod, REML = FALSE)
m26_re <- lmer(yday ~ temp_90 + stream + year + mean_elevation + (1 | COMID), data = df_mod, REML = FALSE)

Adding a random intercept for COMID significantly improved model performance, reducing AIC by >2,700 points compared to the additive fixed-effect model (m16 vs. m16_re; Table 15). This confirms the necessity of accounting for repeated measures and reach-specific baseline differences in spawn timing.

Adding mean elevation as a fixed effect (m26_re) yielded a further AIC improvement of 126 points over m16_re. However, this is substantially smaller than the apparent benefit observed in the fixed-effects-only comparison (296 AIC points; m26 vs. m16), suggesting that much of elevation’s explanatory power is absorbed by the random intercepts.

Table 15: Model performace for base random intercept models.
Name Model RMSE R2_conditional R2_marginal ICC R2 R2_adjusted AIC_wt BIC_wt Performance_Score
m26_re lmerMod 2.047765 0.9572783 0.7590988 0.8226590 NA NA 1 1 0.9997827
m16_re lmerMod 2.046763 0.9818897 0.7051542 0.9385771 NA NA 0 0 0.3333333
m26 lm 3.411572 NA NA NA 0.8332567 0.8325904 0 0 0.0374426
m16 lm 3.584278 NA NA NA 0.8159471 0.8152732 0 0 0.0000000

Moreover, the parameter estimate for elevation remained positive (Table 16) and counter to biological expectations (i.e., higher elevation reaches spawning later), and the associated prediction plot (Figure 20C) showed a weak and potentially misleading relationship.

Table 16: Parameter estimates for base random intercept models.
  m16 m26 m16_re: random intercepts m26_re: random intercepts + elevation
Predictors Estimates CI p Estimates CI p Estimates CI p Estimates CI p
(Intercept) 237.80 237.42 – 238.18 <0.001 233.73 233.16 – 234.31 <0.001 234.23 229.39 – 239.06 <0.001 223.56 220.51 – 226.62 <0.001
temp 90 7.18 7.01 – 7.36 <0.001 9.41 9.11 – 9.70 <0.001 15.30 15.03 – 15.56 <0.001 15.34 15.08 – 15.60 <0.001
stream [Beaver] 8.11 7.47 – 8.74 <0.001 9.68 9.05 – 10.31 <0.001 20.38 12.62 – 28.14 <0.001 15.53 11.18 – 19.89 <0.001
stream [Big] 0.86 0.42 – 1.30 <0.001 10.96 9.77 – 12.16 <0.001 -2.30 -8.27 – 3.67 0.450 34.73 28.84 – 40.62 <0.001
stream [Camas] 2.15 1.64 – 2.66 <0.001 8.86 7.97 – 9.75 <0.001 3.42 -2.87 – 9.70 0.287 25.48 20.91 – 30.05 <0.001
stream [Elk] -0.25 -0.74 – 0.23 0.306 1.20 0.71 – 1.69 <0.001 11.02 2.89 – 19.14 0.008 9.10 4.58 – 13.61 <0.001
stream [Loon] 5.08 4.52 – 5.65 <0.001 11.90 10.97 – 12.83 <0.001 0.19 -5.90 – 6.27 0.952 26.35 21.53 – 31.18 <0.001
stream [Marsh] 6.17 5.68 – 6.66 <0.001 4.97 4.49 – 5.45 <0.001 6.89 0.31 – 13.47 0.040 4.66 0.97 – 8.34 0.013
stream [Sulphur] 4.95 4.32 – 5.58 <0.001 13.17 12.07 – 14.26 <0.001 20.86 14.00 – 27.71 <0.001 33.47 29.25 – 37.69 <0.001
year [2003] -2.84 -3.15 – -2.52 <0.001 -3.78 -4.09 – -3.46 <0.001 -5.62 -5.82 – -5.41 <0.001 -5.63 -5.84 – -5.42 <0.001
year [2004] 1.73 1.31 – 2.15 <0.001 2.15 1.75 – 2.55 <0.001 2.87 2.62 – 3.13 <0.001 2.89 2.63 – 3.14 <0.001
year [2005] 3.91 3.38 – 4.45 <0.001 3.94 3.42 – 4.45 <0.001 4.28 3.95 – 4.60 <0.001 4.28 3.95 – 4.60 <0.001
mean elevation 4.64 4.13 – 5.16 <0.001 14.89 12.93 – 16.85 <0.001
Random Effects
σ2     4.34 4.34
τ00     66.27 COMID 20.12 COMID
ICC     0.94 0.82
N     104 COMID 104 COMID
Observations 3016 3016 3016 3016
R2 / R2 adjusted 0.816 / 0.815 0.833 / 0.833 0.705 / 0.982 0.759 / 0.957
Predicted relationship (red line) between spawn date (`yday`) and (A) `temp_90`, and (B) `mean_elevation` for the random intercept model (`m26_re`).

Figure 20: Predicted relationship (red line) between spawn date (yday) and (A) temp_90, and (B) mean_elevation for the random intercept model (m26_re).

This, combined with collinearity concerns and limited interpretability, led us to exclude mean elevation from further models. Subsequent model refinement focused on fixed effects of temperature, stream, and year, with COMID modeled as a random intercept or slope to capture spatial variation in thermal response.

4.4.2 Quadratic Term for Temperature

We next tested whether a nonlinear temperature response improved model fit by adding a quadratic term for temp_90 to the random intercept model (m16_re).

m16_re_quad <- lmer(
  yday ~ temp_90 + I(temp_90^2) + stream + year + (1 | COMID), 
  data = df_mod, REML = FALSE
  )

This resulted in a lower AIC (ΔAIC = 91.7; Table 17), indicating substantial support for the quadratic formulation (m16_re_quad). The fitted curve reflects biologically plausible curvature — a decelerating response at higher temperatures — consistent with expectations for thermal constraints on salmonid spawning (Figure 21C). We therefore retained the quadratic term for temperature in all subsequent models.

Table 17: Model selection for m16_re vs. m16_re_quad.
Model df AIC delta
m16_re_quad 15 13473.12 0.00000
m16_re 14 13564.49 91.36939
Predicted relationship (red line) between spawn date (`yday`) and `temp_90` for the variations on `m16`.

Figure 21: Predicted relationship (red line) between spawn date (yday) and temp_90 for the variations on m16.

4.4.3 Random slopes for temp_90

To account for variation in temperature sensitivity across sites, we extended our top random intercept model (m16_re_quad) by allowing COMID-specific random slopes for temperature (m16_re_quad_rs). This model had the same fixed effects structure but included (1 + temp_90 | COMID).

m16_re_quad <- lmer(
  yday ~ temp_90 + I(temp_90^2) + stream + year + (1 | COMID),
  data = df_mod, REML = TRUE
  )
m16_re_quad_rs <- lmer(
  yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID), 
  data = df_mod, REML = TRUE
  )

The addition of random slopes substantially improved model fit (ΔAIC = 510; Table 18), and diagnostic plots confirmed adequate model behavior.

Adding random slopes increased model flexibility, which redistributed explained variance from fixed to random effects. As a result, marginal R² (variance explained by fixed effects) decreased from 0.726 to 0.701, while conditional R² (total variance explained) remained high at 0.98 in both models (Table 19. This tradeoff reflects a more realistic partitioning of variance, acknowledging that some differences in thermal sensitivity are site-specific and better explained by random effects.

Table 18: Model selection for m16_re_quad vs. m16_re_quad_rs.
Model df AIC delta
m16_re_quad_rs 17 12948.76 0.0000
m16_re_quad 15 13459.05 510.2834
Table 19: Parameter estimates for m16_re_quad and m16_re_quad_rs.
  m16_re_quad m16_re_quad_rs
Predictors Estimates CI p Estimates CI p
(Intercept) 234.77 230.24 – 239.30 <0.001 235.09 230.95 – 239.22 <0.001
temp 90 14.74 14.46 – 15.02 <0.001 13.90 13.06 – 14.74 <0.001
temp 90^2 -0.65 -0.78 – -0.52 <0.001 -0.85 -1.15 – -0.55 <0.001
stream [Beaver] 19.48 12.22 – 26.75 <0.001 17.41 10.62 – 24.19 <0.001
stream [Big] -0.93 -6.53 – 4.67 0.745 -0.19 -5.37 – 5.00 0.944
stream [Camas] 3.65 -2.23 – 9.54 0.224 2.30 -3.08 – 7.69 0.402
stream [Elk] 10.52 2.91 – 18.13 0.007 11.97 4.88 – 19.06 0.001
stream [Loon] 1.07 -4.63 – 6.78 0.713 2.80 -2.47 – 8.06 0.298
stream [Marsh] 6.72 0.56 – 12.88 0.033 7.93 2.30 – 13.56 0.006
stream [Sulphur] 20.41 13.99 – 26.83 <0.001 14.69 8.69 – 20.69 <0.001
year [2003] -5.14 -5.36 – -4.91 <0.001 -5.00 -5.23 – -4.77 <0.001
year [2004] 2.76 2.51 – 3.02 <0.001 2.77 2.53 – 3.01 <0.001
year [2005] 4.18 3.86 – 4.51 <0.001 3.68 3.38 – 3.98 <0.001
Random Effects
σ2 4.24 3.36
τ00 58.03 COMID 49.73 COMID
τ11   12.63 COMID.temp_90
ρ01   -0.23 COMID
ICC 0.93 0.95
N 104 COMID 104 COMID
Observations 3016 3016
Marginal R2 / Conditional R2 0.714 / 0.981 0.698 / 0.985

4.4.4 Partitioning Variation Between Nonlinearity and Heterogeneity

Here, we compare best random slope model with and without a quadratic effect. This tests: does random slope replace the need for quadratic, or do both help? We must refit with ML to compare models with different fixed effects.

m16_re_quad_rs <- lmer(
  yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID), 
  data = df_mod, REML = FALSE
  )
m16_re_linear_rs <- lmer(
  yday ~ temp_90 + stream + year + (1 + temp_90 | COMID),
  data = df_mod, REML = FALSE
  )
Table 20: Model selection for m16_re_quad_rs vs. m16_re_linear_rs.
Model df AIC delta
m16_re_quad_rs 17 12965.97 0.00000
m16_re_linear_rs 16 12986.41 20.44148
Model predictions (red lines) for `m16_re_quad_rs` and `m16_re_linear_rs`.

Figure 22: Model predictions (red lines) for m16_re_quad_rs and m16_re_linear_rs.

The quadratic term improves fit modestly (ΔAIC = 20.4) while keeping the structure stable. Most of the model’s explanatory power is coming from the random slopes, not the curvature of temperature. But the quadratic does meaningfully refine the relationship — especially at the tails of the temperature gradient (Figure 22) — without overfitting or destabilizing the model. This model is supported. This structure offered a strong balance between flexibility and interpretability and was retained for subsequent analysis.

4.5 Interactions

We now have built a strong foundation showing that COMID-level variation captures spatial structure in the data (m16_re_quad_rs).

  • Demonstrated support for a nonlinear (quadratic) thermal response.
  • Incorporated year and stream as fixed effects for interpretability across broad groups.

So now it does make sense to check interactions, with this rationale:

Even though random slopes already absorb much of the temperature-related heterogeneity, interactions can help us test whether average responses differ systematically across stream or year, beyond what’s explained by the random effects.

Logical interactions to test:

  • temp_90 * stream – Does the average thermal slope vary among streams?
  • temp_90 * year – Does thermal sensitivity vary across years (e.g., wetter vs. drier)?

We will now test interactions and compare them to m16_re_quad_rs.

m201 <- lmer(yday ~ temp_90 * stream + I(temp_90^2) + year + (1 + temp_90 | COMID), data = df_mod, REML = FALSE)
m202 <- lmer(yday ~ temp_90 * year + I(temp_90^2) + stream + (1 + temp_90 | COMID), data = df_mod, REML = FALSE)

Table 21 shows that m201 lowers AIC by 18, while m202 lowers AIC by 1397, a massive improvement.

Table 22 shows that the interaction terms in m201 are mostly non-significant, and the model R² values are nearly identical to m16_re_quad_rs.

Table 21: Model selection for m16_re_quad_rs vs. m201 and m202.
Model df AIC delta
m202 20 11568.63 0.000
m201 24 12948.51 1379.886
m16_re_quad_rs 17 12965.97 1397.340
Table 22: Parameter estimates for interaction models.
  m16_re_quad_rs m201: temp_90 * stream m202: temp_90 * year
Predictors Estimates CI p Estimates CI p Estimates CI p
(Intercept) 235.12 231.18 – 239.05 <0.001 234.55 230.27 – 238.83 <0.001 232.86 226.46 – 239.25 <0.001
temp 90 13.85 13.02 – 14.68 <0.001 14.99 13.22 – 16.76 <0.001 18.84 17.87 – 19.80 <0.001
temp 90^2 -0.86 -1.16 – -0.56 <0.001 -1.12 -1.43 – -0.82 <0.001 1.53 1.23 – 1.83 <0.001
stream [Beaver] 17.37 10.90 – 23.84 <0.001 17.93 11.04 – 24.83 <0.001 21.60 11.35 – 31.85 <0.001
stream [Big] -0.17 -5.11 – 4.78 0.947 -1.07 -6.48 – 4.33 0.697 -3.78 -11.78 – 4.23 0.355
stream [Camas] 2.29 -2.84 – 7.42 0.381 2.00 -3.59 – 7.59 0.483 1.82 -6.51 – 10.14 0.668
stream [Elk] 11.88 5.12 – 18.64 0.001 11.74 4.53 – 18.95 0.001 15.89 5.13 – 26.64 0.004
stream [Loon] 2.85 -2.17 – 7.87 0.266 3.43 -2.21 – 9.06 0.233 1.71 -6.48 – 9.89 0.683
stream [Marsh] 7.91 2.55 – 13.27 0.004 9.15 3.35 – 14.96 0.002 9.50 0.82 – 18.18 0.032
stream [Sulphur] 14.69 8.97 – 20.42 <0.001 15.37 9.26 – 21.49 <0.001 24.50 15.44 – 33.56 <0.001
year [2003] -4.99 -5.22 – -4.76 <0.001 -4.96 -5.19 – -4.73 <0.001 -7.03 -7.23 – -6.83 <0.001
year [2004] 2.76 2.52 – 3.00 <0.001 2.78 2.53 – 3.02 <0.001 2.96 2.77 – 3.15 <0.001
year [2005] 3.68 3.38 – 3.98 <0.001 3.69 3.39 – 4.00 <0.001 3.75 3.51 – 3.98 <0.001
temp 90 × stream [Beaver] -2.81 -5.85 – 0.23 0.070
temp 90 × stream [Big] 0.83 -1.40 – 3.06 0.466
temp 90 × stream [Camas] 0.99 -1.42 – 3.39 0.421
temp 90 × stream [Elk] 0.68 -2.30 – 3.66 0.656
temp 90 × stream [Loon] -1.29 -3.87 – 1.28 0.325
temp 90 × stream [Marsh] -3.96 -6.48 – -1.44 0.002
temp 90 × stream
[Sulphur] 		-5.05 -7.75 – -2.35 <0.001
temp 90 × year [2003] -3.89 -4.08 – -3.70 <0.001
temp 90 × year [2004] 0.63 0.40 – 0.87 <0.001
temp 90 × year [2005] -1.29 -1.65 – -0.93 <0.001
Random Effects
σ2 3.36 3.37 1.96
τ00 45.06 COMID 50.42 COMID 114.83 COMID
τ11 12.18 COMID.temp_90 6.52 COMID.temp_90 17.85 COMID.temp_90
ρ01 -0.23 COMID -0.35 COMID 0.00 COMID
ICC 0.94 0.94 0.99
N 104 COMID 104 COMID 104 COMID
Observations 3016 3016 3016
Marginal R2 / Conditional R2 0.713 / 0.984 0.736 / 0.985 0.620 / 0.994

For m202, while the interactions terms are significant, the Marginal R² value drop by ~10%.

Figure 23 shows that the predicted effects of temperature are implausible, with a positive quadratic effect of temperature. This suggests that the model is overfitting or confounding the interaction structure.

Predicted spawn timing.

Figure 23: Predicted spawn timing.

4.5.1 Summary of interactions

To evaluate whether fixed-effect interactions improve model performance beyond what is captured by random slopes, we tested two candidate models: one with a temp_90 × stream interaction (m201) and one with a temp_90 × year interaction (m202), comparing them against our baseline random slope model with a quadratic temperature effect (m16_re_quad_rs).

Model m202 showed a large AIC improvement (ΔAIC = -1397), but inspection of its predicted effects revealed implausible curvature—specifically, a biologically unlikely inverted quadratic effect of temperature. This suggests overfitting or confounding in the interaction structure.

In contrast, m201 showed moderate AIC improvement over the baseline model (ΔAIC = 18), with predictions that remained biologically realistic. However, nearly all interaction terms in m201 were non-significant except one, and model R² values and parameter estimates remained essentially unchanged.

These results indicate that most stream-specific variation in temperature responses is already accounted for by random slopes. Thus, the added complexity of fixed-effect interactions does not yield substantial inferential gains and is not justified. We therefore retained m16_re_quad_rs as our final model.

5 Model interpretation

5.1 Final Model

The final model included a linear and quadratic effect of temperature (temp_90, I(temp_90^2)), additive fixed effects for stream and year, and a random intercept and slope for temperature by COMID:

mod_final <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID), 
  data = df_mod, REML = TRUE)

This model balances explanatory power, biological realism, and parsimony. It captures both general patterns in spawn timing (via fixed effects) and local deviations in temperature response (via random slopes), and will serve as the basis for diagnostics, prediction, and ecological interpretation.

5.2 Model summary and fit

Parameter estimates (Table 23) indicate a significant nonlinear effect of temperature on spawn timing, with the quadratic term (–0.85) confirming a concave-down relationship—i.e., spawn timing advances with increasing temperature, but levels off at high temperatures.

Most stream and year effects were significant, capturing expected spatiotemporal structure in spawn timing. Notably, Big, Camas, and Loon were not significantly different from the reference level (Bear Valley).

The random effects structure reveals substantial variation across COMIDs. The standard deviation for random intercepts (√τ₀₀ = 7.05) and random slopes for temp_90 (√τ₁₁ = 3.55) indicates strong spatial heterogeneity in both baseline timing and temperature sensitivity (Table 23). The intraclass correlation (ICC) was high (0.95), reflecting strong grouping structure at the COMID level (Table 24).

Model fit was excellent: the marginal R² (variance explained by fixed effects) was 0.698, and the conditional R² (fixed + random effects) was 0.985 (Table 24). This suggests that the majority of explanatory power is derived from spatially varying thermal responses captured by the random slopes, with additional structure provided by the fixed effects.

Table 23: Parameter estimates for mod_final.
Parameter Coefficient SE CI CI_low CI_high t df_error p Effects Group
(Intercept) 235.09 2.11 0.95 230.95 239.22 111.50 2999 0.00 fixed
temp_90 13.90 0.43 0.95 13.06 14.74 32.50 2999 0.00 fixed
I(temp_90^2) -0.85 0.15 0.95 -1.15 -0.55 -5.54 2999 0.00 fixed
streamBeaver 17.41 3.46 0.95 10.62 24.19 5.03 2999 0.00 fixed
streamBig -0.19 2.65 0.95 -5.37 5.00 -0.07 2999 0.94 fixed
streamCamas 2.30 2.75 0.95 -3.08 7.69 0.84 2999 0.40 fixed
streamElk 11.97 3.61 0.95 4.88 19.06 3.31 2999 0.00 fixed
streamLoon 2.80 2.68 0.95 -2.47 8.06 1.04 2999 0.30 fixed
streamMarsh 7.93 2.87 0.95 2.30 13.56 2.76 2999 0.01 fixed
streamSulphur 14.69 3.06 0.95 8.69 20.69 4.80 2999 0.00 fixed
year2003 -5.00 0.12 0.95 -5.23 -4.77 -42.79 2999 0.00 fixed
year2004 2.77 0.12 0.95 2.53 3.01 22.57 2999 0.00 fixed
year2005 3.68 0.15 0.95 3.38 3.98 24.00 2999 0.00 fixed
SD (Intercept) 7.05 NA 0.95 NA NA NA NA NA random COMID
SD (temp_90) 3.55 NA 0.95 NA NA NA NA NA random COMID
Cor (Intercept~temp_90) -0.23 NA 0.95 NA NA NA NA NA random COMID
SD (Observations) 1.83 NA 0.95 NA NA NA NA NA random Residual
Table 24: Model performance for mod_final.
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
12948.76 12948.97 13050.96 0.9845502 0.6979933 0.9488428 1.780765 1.833613

5.3 Residual diagnostics

Model diagnostics for the final model were generally strong (Figure 24). The posterior predictive check shows excellent agreement between the observed and model-predicted distributions of spawn timing, with overlapping density curves and no major deviations.

Plots of linearity and homoscedasticity indicate acceptable model performance. While the residual vs. fitted plot shows a slight trend, it does not appear strong enough to invalidate model assumptions. The spread of residuals is approximately constant across fitted values, though there is minor funneling at the lower end, likely reflecting skew in early spawn dates.

Influential observations are limited. A few data points exceed standard influence thresholds (|standardized residual| > 2 and high leverage), but none are extreme enough to warrant removal, and their leverage is modest. The model appears robust to outliers.

Collinearity is low: variance inflation factors (VIFs) for all fixed effects are well below the conservative threshold of 5, suggesting little concern about multicollinearity.

The normal Q-Q plot of residuals shows slight right-skew and some heavy tails, but the distribution is reasonably close to normal. Similarly, random effect Q-Q plots for intercepts and slopes show mostly linear trends with slight deviation at the tails, indicating acceptable assumptions for the mixed-effects structure.

Overall, the model shows no violations of key assumptions and is suitable for inference and prediction.

Residual diagnostics for `mod_final`.

Figure 24: Residual diagnostics for mod_final.

5.4 Population-level effects

5.4.1 Predictions against original data

Model predictions closely matched observed spawn timing, with predicted values aligning well along the 1:1 line (Figure 25). This supports strong overall model fit.

Predicted vs. observed spawn timing for Chinook Salmon across all years and streams. The dashed line shows the 1:1 line. Points represent individual redd observations.

Figure 25: Predicted vs. observed spawn timing for Chinook Salmon across all years and streams. The dashed line shows the 1:1 line. Points represent individual redd observations.

5.4.2 Marginal means and contrasts of yday at each factor level

We estimated marginal mean of yday at each factor level, averaging over the random effects, to provide an overall estimate of the effect in the population (Figure 26).

Predicted mean spawn dates by stream and year (A), stream (B), and year (C) from the final mixed-effects model (yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID)). Colored points with horizontal lines (A) and black points with black lines (B, C) represent eatimated marginal means and 95% confidence intervals. Background boxplots in panels B and C show the distribution of observed redd counts by group, with individual data points in grey. The modeled predictions represent marginal means accounting for fixed effects and averaged over random effects.

Figure 26: Predicted mean spawn dates by stream and year (A), stream (B), and year (C) from the final mixed-effects model (yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID)). Colored points with horizontal lines (A) and black points with black lines (B, C) represent eatimated marginal means and 95% confidence intervals. Background boxplots in panels B and C show the distribution of observed redd counts by group, with individual data points in grey. The modeled predictions represent marginal means accounting for fixed effects and averaged over random effects.

Significant differences in spawn timing were observed between many stream pairs (Table 25), particularly involving Loon (later spawning) and Sulphur (earlier spawning). For example, fish in Loon spawned significantly later than in Bear Valley, Camas, and Elk, while Sulphur exhibited significantly earlier timing than all other streams except Elk. These patterns reflect spatial heterogeneity in temperature and elevation across streams that is not fully captured by fixed effects alone.

Table 25: Pairwise contrasts among stream effects on predicted spawn timing. Contrasts represent estimated differences in predicted spawn day of year between streams from the final model. Positive values indicate later predicted spawn timing in the first stream compared to the second. P-values are uncorrected.
Level1 Level2 Difference SE CI_low CI_high t df p
Beaver Bear Valley -1.43 3.47 -8.23 5.38 -0.41 2999 0.68
Big Bear Valley -2.66 2.66 -7.88 2.56 -1.00 2999 0.32
Camas Bear Valley 0.63 2.75 -4.77 6.03 0.23 2999 0.82
Elk Bear Valley -6.60 3.62 -13.71 0.50 -1.82 2999 0.07
Loon Bear Valley 8.90 2.68 3.64 14.15 3.32 2999 0.00
Marsh Bear Valley 6.72 2.87 1.08 12.35 2.34 2999 0.02
Sulphur Bear Valley -9.47 3.16 -15.66 -3.28 -3.00 2999 0.00
Big Beaver -1.23 3.18 -7.46 4.99 -0.39 2999 0.70
Camas Beaver 2.06 3.27 -4.35 8.46 0.63 2999 0.53
Elk Beaver -5.18 4.01 -13.04 2.69 -1.29 2999 0.20
Loon Beaver 10.33 3.24 3.98 16.67 3.19 2999 0.00
Marsh Beaver 8.14 3.41 1.46 14.82 2.39 2999 0.02
Sulphur Beaver -8.04 3.54 -14.98 -1.10 -2.27 2999 0.02
Camas Big 3.29 2.40 -1.42 8.00 1.37 2999 0.17
Elk Big -3.94 3.35 -10.51 2.62 -1.18 2999 0.24
Loon Big 11.56 2.35 6.95 16.16 4.92 2999 0.00
Marsh Big 9.38 2.57 4.33 14.42 3.64 2999 0.00
Sulphur Big -6.81 2.78 -12.26 -1.36 -2.45 2999 0.01
Elk Camas -7.23 3.43 -13.97 -0.50 -2.11 2999 0.04
Loon Camas 8.27 2.45 3.47 13.06 3.38 2999 0.00
Marsh Camas 6.09 2.66 0.87 11.30 2.29 2999 0.02
Sulphur Camas -10.10 2.91 -15.80 -4.40 -3.47 2999 0.00
Loon Elk 15.50 3.40 8.84 22.17 4.56 2999 0.00
Marsh Elk 13.32 3.56 6.34 20.30 3.74 2999 0.00
Sulphur Elk -2.87 3.71 -10.14 4.41 -0.77 2999 0.44
Marsh Loon -2.18 2.57 -7.23 2.87 -0.85 2999 0.40
Sulphur Loon -18.37 2.91 -24.07 -12.66 -6.31 2999 0.00
Sulphur Marsh -16.19 3.10 -22.27 -10.10 -5.22 2999 0.00

There was a clear trend toward later spawning over the four-year period (Table 25). Spawning in 2005 occurred significantly later than in all previous years. Differences between 2002 and 2003 were not statistically significant, but later years (2004 and especially 2005) were associated with a progressive delay in mean spawn timing. This temporal shift likely reflects interannual variability in temperature and flow conditions.

Table 26: Pairwise contrasts among year effects on predicted spawn timing. Contrasts represent estimated differences in predicted spawn day of year between years from the final model. Positive values indicate later spawning in the first year compared to the second. P-values are uncorrected.
Level1 Level2 Difference SE CI_low CI_high t df p
2003 2002 0.97 0.57 -0.15 2.10 1.70 2999 0.09
2004 2002 2.35 0.65 1.07 3.62 3.61 2999 0.00
2005 2002 6.18 0.61 4.98 7.37 10.17 2999 0.00
2004 2003 1.37 0.61 0.18 2.56 2.26 2999 0.02
2005 2003 5.20 0.71 3.80 6.60 7.28 2999 0.00
2005 2004 3.83 0.87 2.11 5.54 4.38 2999 0.00

5.4.3 Estimating response vs. relation

To visualize the model’s predictions across the temperature gradient, we estimated the relationship between spawn timing and 90-day pre-spawn mean temperature (temp_90) for different groupings: overall, by stream, and by year. This approach allows us to assess both general trends and context-specific responses.

Stream-specific predictions (see plot suggestion below) show a consistent pattern: spawn timing increases nonlinearly with 90-day mean stream temperature, leveling off at high temperatures. This plateau is consistent with biological expectations, as spawning may be constrained by environmental or physiological thresholds. Stream-to-stream variation in predicted timing reflects both fixed stream effects and COMID-specific random intercepts and slopes.

Year-specific predictions similarly show consistent thermal responses across years, with modest offsets in average spawn timing due to year effects. These predictions further validate the model’s generalizability and temporal consistency.

In addition, a combined stream-by-year plot (e.g., facet grid) confirms that the final model captures heterogeneity in both space and time without overfitting. Lines track the raw data closely across groups, particularly in mid-range temperatures where most observations are concentrated.

Together, these plots indicate that the final model effectively captures both nonlinear temperature effects and spatial variation in thermal sensitivity, while maintaining interpretability and predictive strength.

When stratified by stream and year, predictions captured both the nonlinear temperature response and variation in baseline spawn timing across sites and years (Figure 27). The curvature was consistent across years, while intercept shifts reflected known spatial patterns (e.g., later spawning in warmer, downstream reaches). These results confirm that the model accurately describes both average and context-specific phenological responses to temperature.

Predicted relationship between spawn timing and 90-day pre-spawn mean temperature by stream and year. Lines represent model predictions from the final mixed-effects model. Colored points show observed redd timing, shaded ribbons represent 95% confidence intervals for predictions.

Figure 27: Predicted relationship between spawn timing and 90-day pre-spawn mean temperature by stream and year. Lines represent model predictions from the final mixed-effects model. Colored points show observed redd timing, shaded ribbons represent 95% confidence intervals for predictions.

5.5 Group-level effects (deviations from fixed effects)

Random intercepts and slopes varied considerably among COMIDs, reflecting spatial heterogeneity in both average spawn timing and thermal sensitivity (Figure 28A). We found considerable spread in intercepts, reflecting differences in average spawn timing between reaches. The random slopes for temp_90 likewise varied meaningfully across COMIDs, indicating that temperature–spawn timing relationships are not constant across space.

Sites with earlier average spawn timing (lower intercepts) generally exhibited stronger positive responses to temperature (higher slopes), while later-spawning sites tended to show weaker temperature effects, a pattern also evident in the weak negative correlation between intercepts and slopes (r = -0.2; Figure 28B). This indicates that reaches with earlier average spawn timing (i.e., negative intercepts) tend to exhibit stronger temperature sensitivity (i.e., steeper positive slopes), whereas later-spawning reaches show weaker responses to temperature. Biologically, this suggests that early-spawning populations are more phenologically plastic and adjust their timing more closely to thermal cues, while late-spawning populations may be constrained by other factors—such as photoperiod, migration fatigue, or compressed spawning windows—resulting in diminished thermal responsiveness. These findings highlight spatial heterogeneity in phenological flexibility, with potential implications for how different populations may respond to climate change.

(A) COMID-specific random parameter estimates for intercepts (left) and slopes (right). Points represent best linear unbiased predictions (BLUPs) from the final model, with horizontal bars indicating ±1.96 standard errors. (B) Correlation between random intercepts and slopes for 90-day temperature across COMIDs. Each point represents a stream reach (COMID).

Figure 28: (A) COMID-specific random parameter estimates for intercepts (left) and slopes (right). Points represent best linear unbiased predictions (BLUPs) from the final model, with horizontal bars indicating ±1.96 standard errors. (B) Correlation between random intercepts and slopes for 90-day temperature across COMIDs. Each point represents a stream reach (COMID).

When grouped by stream, Bear Valley and Big Creek exhibited early average spawn timing and high thermal sensitivity, while Sulphur and Marsh Creeks had later average timing with flatter temperature responses (Figure 29).

Boxplots of random intercepts and slopes for 90-day pre-spawn temperature by stream. Each box represents the distribution of best linear unbiased predictions (BLUPs) for a COMID's random intercept (average spawn timing) or slope (thermal sensitivity). Bars are colored by stream.

Figure 29: Boxplots of random intercepts and slopes for 90-day pre-spawn temperature by stream. Each box represents the distribution of best linear unbiased predictions (BLUPs) for a COMID’s random intercept (average spawn timing) or slope (thermal sensitivity). Bars are colored by stream.

However, when examing individual random effects for each COMID and stream, we observed considerable variation in both intercepts and slopes within streams (Figure 30). For example, Bear Valley and Big Creek had some of the earliest average spawn timings, but also exhibited a wide range of thermal sensitivities. In contrast, Sulphur Creek had later average spawn timing but also showed considerable variability in its response to temperature.

Random intercepts and slopes for 90-day pre-spawn temperature by stream. Each bar represents the best linear unbiased prediction (BLUP) for a COMID's random intercept (average spawn timing) or slope (thermal sensitivity). Bars are colored by stream.

Figure 30: Random intercepts and slopes for 90-day pre-spawn temperature by stream. Each bar represents the best linear unbiased prediction (BLUP) for a COMID’s random intercept (average spawn timing) or slope (thermal sensitivity). Bars are colored by stream.

Variation in temperature–spawn timing relationships across stream reaches (COMIDs) as estimated from the final mixed-effects model is shown in Figure 31). While the overall relationship is positive and nonlinear, individual COMID slopes and intercepts vary considerably. Some reaches show steeper increases in spawn timing with temperature (i.e., stronger thermal sensitivity), while others are relatively flat, indicating a weaker or more buffered response. Grouping by stream shows that some streams (e.g., Bear Valley, Marsh) exhibit tightly clustered trajectories, while others (e.g., Camas, Big) show more divergence. This variation likely reflects fine-scale differences in local hydrology, geomorphology, or biological factors that influence how fish respond to thermal cues within stream systems. The consistency of the overall trend, despite local heterogeneity, supports the biological relevance of temperature in structuring spawn timing.

Predicted spawn timing by 90-day pre-spawn temperature and COMID. Each line represents the predicted spawn timing for a specific COMID, colored by stream. The black line and shaded ribbon represents the 95% confidence interval for the population-level predictions.

Figure 31: Predicted spawn timing by 90-day pre-spawn temperature and COMID. Each line represents the predicted spawn timing for a specific COMID, colored by stream. The black line and shaded ribbon represents the 95% confidence interval for the population-level predictions.

5.5.1 Elevation effects embedded in random structure

TO DISCUSS

Although mean elevation was excluded as a fixed effect due to collinearity and inconsistent directionality, plots of random effects against elevation reveal underlying spatial structure that elevation helps to explain (Figure 32).

In panel A, there is a clear positive relationship between elevation and the random intercepts for some streams (e.g., Big), indicating that higher elevation sites within Big Creek tend to have later average spawn timing compared to the global (population) average.

In panel B, random slopes (i.e., thermal sensitivity) show more idiosyncratic patterns: for some streams (e.g., Camas, Marsh), thermal sensitivity appears to decrease with elevation, suggesting fish at higher elevations may respond less strongly to temperature variability.

However, these relationships are inconsistent across streams, supporting the decision to capture this spatial heterogeneity via COMID-level random effects rather than forcing a global fixed elevation term.

Relationship between `mean_elevation` and (A) random intercepts and (B) slopes for 90-day pre-spawn temperature, as well as `mean_elevation` and (C) spawn date (`yday`) and (D) `temp_90` . Points and lines are colored by stream, and solid lines represent linear fits.

Figure 32: Relationship between mean_elevation and (A) random intercepts and (B) slopes for 90-day pre-spawn temperature, as well as mean_elevation and (C) spawn date (yday) and (D) temp_90 . Points and lines are colored by stream, and solid lines represent linear fits.